1. Section 3: Uncertainty in Data
Measurements contain uncertainties that affect how a calculated result is presented. K
What I Know W
What I Want to Find Out L
What I Learned

2. Essential Questions How do accuracy and precision compare?
How can the accuracy of data be described using error and percent error?
What are the rules for significant figures and how can they be used to express uncertainty in measured and calculated values?
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Uncertainty in Data

3. Vocabulary Review New experiment accuracy precision error
percent error
significant figure
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Uncertainty in Data

4. Accuracy and Precision
Accuracy refers to how close a measured value is to an accepted value.
Precision refers to how close a series of measurements are to one another.
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Uncertainty in Data

5. Add link to concepts in motion animation from page 47 here.
Precision and Accuracy
Concepts in Motion
FPO
Add link to concepts in motion animation from page 47 here.
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Uncertainty in Data

6. Accuracy and Precision
Error is defined as the difference between an experimental value and an accepted value.
a: These trial values are the most precise
b: This average is the most accurate
Uncertainty in Data
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7. Accuracy and Precision
The error equation is error = experimental value – accepted value.
Percent error expresses error as a percentage of the accepted value.
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Uncertainty in Data

8. CALCULATING PERCENT ERROR
SOLVE FOR THE UNKNOWN
State the percent error equation.
percent error = |error| accepted value ×100
Substitute error = −0.05 g/cm3, and solve.
percent error = |−0.05 g/cm3| 1.59 g/cm3 ×100 =3.14%
Substitute error = 0.01 g/cm3, and solve.
percent error = |0.01 g/cm3| 1.59 g/cm3 ×100 =0.63%
Use with Example Problem 5.
Problem
Use Student A’s density data in Table 3 (from slide 6) to calculate the percent error in each trial. Report your answers to two places after the decimal point.
Response
ANALYZE THE PROBLEM
You are given the errors for a set of density calculations. To calculate percent error, you need to know the accepted value for density, the errors, and the equation for percent error.
KNOWN
UNKNOWN
Accepted value for density = 1.59 g/cm3
Percent errors = ?
Errors: −0.05 g/cm3; 0.01 g/cm3; −0.02 g/cm3
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Uncertainty in Data

9. CALCULATING PERCENT ERROR
EVALUATE THE ANSWER
The percent error is greatest for Trial 1, which had the largest error, and smallest for Trial 2, which was closest to the accepted value.
SOLVE FOR THE UNKNOWN
Substitute error = −0.02 g/cm3, and solve.
percent error = |−0.02 g/cm3| 1.59 g/cm3 ×100 =1.26%
Uncertainty in Data
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10. Significant Figures
Often, precision is limited by the tools available.
Significant figures include all known digits plus one estimated digit.
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Uncertainty in Data

11. Significant Figures Rules for significant figures:
Rule 1: Nonzero numbers are always significant.
Rule 2: Zeros between nonzero numbers are always significant.
Rule 3: All final zeros to the right of the decimal are significant.
Rule 4: Placeholder zeros are not significant. To remove placeholder zeros, rewrite the number in scientific notation.
Rule 5: Counting numbers and defined constants have an infinite number of significant figures.
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Uncertainty in Data

12. Volume = length × width × height
ROUNDING NUMBERS WHEN MULTIPLYING
SOLVE FOR THE UNKNOWN
Calculate the volume, and apply the rules of significant figures and rounding.
State the formula for the volume of a rectangle.
Volume = length × width × height
Substitute values and solve.
Volume = 28.3 cm × 22.2 cm × 3.65 cm = cm3
Round the answer to three significant figures.
Volume = 2290 cm3
Use with Example Problem 8.
Problem
Calculate the volume of a book with the following dimensions: length = 28.3 cm, width = 22.2 cm, height = 3.65 cm.
Response
ANALYZE THE PROBLEM
Volume is calculated by multiplying length, width, and height. Because all of the
measurements have three significant figures, the answer also will.
EVALUATE THE ANSWER
To check if your answer is reasonable, round each measurement to one significant figure and recalculate the volume. Volume = 30 cm × 20 cm × 4 cm = 2400 cm3. Because this value is close to your calculated value of 2290 cm3, it is reasonable to conclude the answer is correct.
KNOWN
UNKNOWN
Length = 28.3 cm
Volume = ? cm3
Width = 22.2 cm
Height = 3.65 cm
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Uncertainty in Data

13. SIGNIFICANT FIGURES Problem Response SOLVE FOR THE UNKNOWN
Rules 1, 2, and 3.
Count all nonzero numbers, zeros between nonzero numbers, and final zeros to the right of the decimal place.
Rule 4.
Ignore zeros that act as placeholders.
a g has five significant figures.
b. 405,000 kg has three significant figures.
Use with Example Problem 6.
Problem
Determine the number of significant figures in the following masses.
a g
b. 405,000 kg
Response
ANALYZE THE PROBLEM
You are given two measured mass values. Apply the appropriate rules to determine the number of significant figures in each value.
EVALUATE THE ANSWER
One way to verify your answers is to write the values in scientific notation:
× 10-4 g and 4.05 × 105 kg. Without the placeholder zeros, it is clear that g has five significant figures and that 405,000 kg has three significant figures.
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Uncertainty in Data

14. Rounding Numbers Calculators are not aware of significant figures.
Answers should not have more significant figures than the original data with the fewest figures, and should be rounded.
Rules for rounding:
Rule 1: If the digit to the right of the last significant figure is less than 5, do not change the last significant figure → 2.53
Rule 2: If the digit to the right of the last significant figure is greater than 5, round up the last significant figure → 2.54
Rule 3: If the digits to the right of the last significant figure are a 5 followed by a nonzero digit, round up the last significant figure → 2.54
Rule 4: If the digits to the right of the last significant figure are a 5 followed by a 0 or no other number at all, look at the last significant figure. If it is odd, round it up; if it is even, do not round up → 2.54
→ 2.52
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Uncertainty in Data

15. Rounding Numbers Addition and subtraction
Round the answer to the same number of decimal places as the original measurement with the fewest decimal places.
Multiplication and division
Round the answer to the same number of significant figures as the original measurement with the fewest significant figures.
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Uncertainty in Data

16. ROUNDING NUMBERS WHEN ADDING
SOLVE FOR THE UNKNOWN
Align the measurements and add the values.
28.0 cm cm cm cm
Round to one place after the decimal; Rule 1 applies.
The answer is 77.2 cm.
Use with Example Problem 7.
Problem
A student measured the length of his lab partners’ shoes. If the lengths are cm, cm, and cm, what is the total length of the shoes?
Response
ANALYZE THE PROBLEM
The three measurements need to be aligned on their decimal points and added. The measurement with the fewest digits after the decimal point is 28.0 cm, with one digit. Thus, the answer must be rounded to only one digit after the decimal point.
EVALUATE THE ANSWER
The answer, 77.2 cm, has the same precision as the least-precise measurement, 28.0 cm.
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Uncertainty in Data

17. Review Essential Questions Vocabulary
How do accuracy and precision compare?
How can the accuracy of data be described using error and percent error?
What are the rules for significant figures and how can they be used to express uncertainty in measured and calculated values?
Vocabulary
accuracy
precision
error
percent error
significant figure
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Uncertainty in Data